No. The engineers of the Internet designed it this way so you have no control over your kids and only they can see what your kids are up to. See also: Google purging location history for anyone who visited an abortion clinic so their dads don't find out.
HTTPS encryption is used on all major platforms, which prevents someone outside from eavesdropping. You would need to use SSL termination which is essentially where you have a device which spoofs the web server certificate (and then all of your devices are set up to accept the spoofed certificate). Without SSL termination, you can't peek into the contents of the HTTPS tunnel. HTTPS prevents, for example, your ISP from seeing your Google search history. Your ISP can see that you went to Google, but not what you searched for. Google wants to keep all that data for themselves.
Potentially you can run something locally on the machine to restrict web access, but that would be up to the software company as to what features. They would most likely route your DNS through their own filtered servers, but a wiley kid with admin access could probably turn that off pretty easily.
Parental controls were a distant afterthought of the Internet and nobody does it well.